February 7

Q1

Solve the congruence
$x^2+x-1\equiv 0 \mod 11$

Q2

For what value of a does the congruence
$x^2+x+a\equiv 0 \mod 11$
have exactly one solution?

Q3

Show that if
$2^n-1$
is prime, then
$n|2^n-2$
Hint: order of an element modulo a prime p divides p-1

Q4

How many elements of order 6 are there modulo 13? List them all.
How many elements of order 5 are there modulo 13?

Q5

How many polynomials
$x^k+a_1x^{k-1}+\ldots+a_k$
of degree k with coefficients 0 and 1 don't have any roots modulo 2?

How many have exactly two roots?

Can such a polynomial have more than two roots?